MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Hi,

let $k/\mathbb{Q}$ be a number field. Assume that $u$ is an algebraic integer such that all $k$-conjugates have modulus $1$. Is $u$ a root of $1$ ?

If $k=\mathbb{Q}$, the answer is YES (this is Kronecker's theorem). I am pretty sure that this result is false if $k$ is an arbitrary number field, but I don't see any obvious counter-example.

Any suggestion ?

Thanks!

share|cite|improve this question
    
If $u\in k$ and $k$ is Galois over $\bf Q$, then this is true. See Lemma 1.6 in Washington, "Introduction to cyclotomic fields". – Damian Rössler Oct 17 '12 at 20:53
    
You should register an account to prevent duplicate identities. – S. Carnahan Nov 2 '12 at 7:36

The answer depends on the number field $k$. Of course, it cannot hold for all fields $k$, for if $u$ is an algebraic integer of modulus $1$ which is not a root of unity (there are plenty of them, see e.g. this MO-link), then set $k=\mathbb Q(u)$, so $u$ is the only $k$-conjugate of $u$.

share|cite|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.